Subloop whose left cosets are pairwise disjoint

Definition

A subloop whose left cosets are pairwise disjoint, or equivalently, a subloop with well-defined left cosets, is a subloop $S$ of an algebra loop $(L,*)$ satisfying the following equivalent conditions:

1. $a*(b*S)=(a*b)*S$ for all $a,b\in L$.
2. The sets $a*S$, for $a\in L$, form a partition of $L$.
3. Given $a,b\in L$, either $a*S=b*S$ or they are pairwise disjoint.
4. The relation $a\sim b\iff a\in b*S$ is an equivalence relation on $L$.

Note that because left cosets partition a group, any subgroup of a group is a subloop whose left cosets are pairwise disjoint.

Relation with other properties

Stronger properties

• Normal subloop

Weaker properties

• Subloop

Related properties

• Subloop whose right cosets are pairwise disjoint